A Note on Quadratic Forms and the u-invariant

Conditions for equality in Anderson’s theorem

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices--one by Childs et al. [J. Mod. Phys. 47, 155 (2000)] and the other one by Chau [Quant. Inf. Comp. 11, 721 (2011)]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators. © 2011 Chau and Lam; licensee Springer.

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $\pi _{1}^{qtop}(X,x_{0})$ and $\pi _{1}^{wh}(X,x_{0})$ , respectively. In particular, we give necessary or sufficient conditions for equality and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the converse of some of these implications do not hold, in general.

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.

This paper presents an analogue of the spectral mapping theorem for the pseudo S-spectra of quaternionic operators. The theorem is formulated using two set inclusions, and necessary and sufficient conditions for equality are identified. The standard mapping theorem appears as a special case of this theorem. A weaker version is also derived with potential applications in various scenarios. The pseudo S-spectra of rank-one and rank-one nilpotent quaternionic operators are explicitly characterized. Finally, the mapping theorem is further demonstrated for specific classes of quaternionic operators, including rank-one nilpotent, normal, self-adjoint, orthogonal projection, and left multiplication operators.

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The $i$-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its $i$-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.

Sur les dimensions des formes quadratiques de hauteur 2

In this article we study a generalization of the notion of Pfister neighbors. An anisotropic quadratic form ’ over a field F of characteristic not 2 is called a quasi-Pfister neighbor when the anisotropic part (’F(’))an is F(’)-similar to an F(⁄)

Formally real fields from a Galois theoretic perspective

A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.

ABSTRACTFrom a normed quadratic space (V,q), we construct a norm on the Clifford algebra C(V,q). We describe the associated graded form of this norm and give a condition for this norm to be a gauge. Then, we apply our results to prove that for a complete discrete valued field, an anisotropic quadratic form q with dimq = 0 mod 8 and nonsplit Clifford algebra cannot be at the same time a transfer of a K-hermitian form with K∕F an inertial quadratic field extension and a transfer of a T-hermitian form with T∕F a ramified quadratic field extension.

Let WF denote the Witt ring of a field F of characteristic ≠2 and let I n F denote the n-th power of the ideal IF of even-dimensional forms in WF. The Arason-Pfister Hauptsatz states that if 0≠ϕ∈I n F is anisotropic then dim ϕ≥ 2 n . Pfister also showed that if ϕ∈I 3 F is anisotropic and dim ϕ>8 then dim ϕ≥12. We extend this result to I 4 F and show that if ϕ∈I 4 F is anisotropic and dim ϕ>16 then dim ϕ≥24 and we provide some results on anisotropic 24-dimensional forms in I 4 F.

A field F F of characteristic ≠ 2 \neq 2 is said to have finite Pythagoras number if there exists an integer m ≥ 1 m\geq 1 such that each nonzero sum of squares in F F can be written as a sum of ≤ m \leq m squares, in which case the Pythagoras number p ( F ) p(F) of F F is defined to be the least such integer. As a consequence of Pfister’s results on the level of fields, p ( F ) p(F) of a nonformally real field F F is always of the form 2 n 2^n or 2 n + 1 2^n+1 , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form 2 n 2^n , 2 n + 1 2^n+1 , and ∞ \infty can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer n ≥ 1 n\geq 1 there exists a formally real field F F with p ( F ) = n p(F)=n . As a refinement, we will show that if n , m ≥ 2 n,m\geq 2 and k ≥ 1 k\geq 1 are integers such that 2 m ≥ 2 k ≥ n 2m\geq 2^{k}\geq n , then there exists a uniquely ordered field F F with p ( F ) = n p(F)=n and u ( F ) = u ~ ( F ) = 2 m u(F)=\tilde {u}(F)=2m (resp. u ( F ) = u ~ ( F ) = ∞ u(F)=\tilde {u}(F)=\infty ), where u u (resp. u ~ \tilde {u} ) denotes the supremum of the dimensions of anisotropic forms over F F which are torsion in the Witt ring of F F (resp. which are indefinite with respect to each ordering on F F ).

Nonstandard quadratic forms over rational function fields